There is a growing interest on the study of continuum parallel robots (CPRs) due to their higher stiffness and better dynamics capacities than serial continuum robots (SCRs). Several works have focused on the computation of their geometrico- and kinemato-static models, that can be sorted into two main categories: (i) models based on the continuous Cosserat equations: They are very accurate but assessing elastic stability with them is tricky; (ii) discretized models: They allow easily checking the elastic stability but they require a large number of elastic variables to be accurate. In this paper, we extend an approach based on assumed strain modes developed for the dynamics of SCRs to the statics of CPRs. This method is able to predict the robot configuration with an excellent accuracy with a very limited number of elastic variables, contrary to other discretization methods. The method is also more than 100 times faster than finite differences for a better prediction accuracy. Finally, it is possible to assess the robot elastic stability by only checking the Hessian of the potential energy as for any discretization method, thus making the analysis of this property simpler than for the continuous Cosserat model. All results are validated through simulations on two case studies.